By plane

thomsonfly has direct flights from the United Kingdom to Kefalonia island (from Birmingham, Bristol, Cardiff,London Gatwick, London Stansted, Manchester).Fly Thomas Cook has direct flights from the United Kingdom to Kefalonia island (from Birmingham,London Gatwick, Manchester, Newcastle).easyJet has direct flight from London Gatwick to Kefalonia island.
Since May 2018 British Airways has a direct flight from London to Kefalonia.

With other companies it will be necessary to make a stopover in Athens.
Several airlines fly from the United Kingdom to Athens including :British AirwaysOlympic Air
If you come from another island in the Ionian Sea, Corfu for example, you could take a flight of Sky Express that connects the islands of the Ionian Sea.

By car

There are several possibilities :
If you don't mind long drives you can go to Bari (Italy), the company
Ventouris Ferries has a direct connection from Bari to Kefalonia.
You can also go to Brindisi (Italy) to take a ferry to Patras,
If you prefer to minimize the route by car, you go to Venice to take a ferry to Patras with
Minoan Lines or
Anek Lines.
When you are in Patras you go to Killini to take a ferry to Kefalonia (89km of road of which 17km on motorway from Patras to Killini then one hour and twenty minutes on the ferry) with
Kefalonian Lines or
Ionian Ferries.
You could even get a ferry from Brindisi to Igoumenitsa (northern west of mainland Greece) with
Ventouris Ferries or From Bari to Igoumenitsa with Superfast FerriesFrom Igoumenitsa you drive along the coast to the island of Lefkada (a bridge connects the island to mainland Greece), then you cross the island to take a ferry in Vassiliki brings you, in one hour, to the north of the island of Cephalonia, Fiskardo. This road is the least expensive and quickest to reach by car the island of Kefalonia. Ventouris Ferries has a direct connection from Bari to Kefalonia.Red Star Ferries has a direct connection from Brindisi to Kefalonia.